Download On Nano β-open sets

Int. Jr. of Engineering, Contemporary Mathematics and Sciences
Vol. 1, No. 2, July-December 2015
Copyright: MUK Publications
ISSN No: 2250-3099
On Nano β-open sets
A.Revathy1 and Gnanambal Ilango2
1
Department of Mathematics, Sri Eshwar College of Engineering, Coimbatore, Tamilnadu, India.
2
Department of Mathematics, Government Arts College, Coimbatore, Tamilnadu, India.
e-mail : 1 [email protected]
ABSTRACT
In this paper we introduce a new class of sets called nano β-open sets in nano topological
spaces. Various forms of nano β-open set corresponding to different cases of approximations
are also derived.
KEYWORDS
Nano topology, nano open sets, nano closed sets, nano interior, nano closure, nano β-open
sets.
2000 Mathematics Subject Classification : 54A05.
1
INTRODUCTION
Abd El Monsef et al. [1] introduced the notion of β-open set in topology, and the equivalent
notion of semi-pre open set was given independently by Andrijevic in [2], and further investigated by Ganster and Andrijevic [3]. The notion of nano topology was introduced by Lellis
Thivagar [4], which was defined in terms of approximations and boundary region of a subset of
an universe using an equivalence relation on it. The subset generated by lower approximations
is characterized by certain objects that will definitely form part of an interest subset, whereas
the upper approximation is characterized by uncertain objects that will possibly form part of
an interest subset. The elements of a nano topological space are called the nano open sets. He
has also defined nano closed sets, nano interior and nano closure. He has introduced the weak
forms of nano open sets namely nano α-open sets, nano semi-open sets, nano pre open sets and
nano regular open sets [5]. In this paper we introduce nano β-open sets. Also various forms of
nano β-open set are studied for various cases of approximations.
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PRELIMINARIES
Definition 2.1. [5]
Let U be a non-empty finite set of objects called the universe and R be an equivalence
relation on U named as the indiscernibility relation. Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair (U, R) is said to be the
approximation space. Let X ⊆ U.
(i) The lower approximation of X with respect to R is the set of all objects, which can be for
certain
classified as X with respect to R and it is denoted by LR (X). That is,LR (X) =
S
{R(x)
: R(x) ⊆ X}, where R(x) denotes the equivalence class determined by x.
x∈U
(ii) The upper approximation of X with respect to R is the set of all objects, which can be
possibly
classified as X with respect to R and it is denoted by UR (X). That is,UR (X) =
S
{R(x)
: R(x) ∩ X 6= ∅}
x∈U
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(iii) The boundary region of X with respect to R is the set of all objects, which can be
classified neither as X nor as not - X with respect to R and it is denoted by BR (X). That
is, BR (X) = UR (X) − LR (X).
Proposition 2.1. [4]
If (U, R) is an approximation space and X, Y ⊆ U, then
(i) LR (X) ⊆ X ⊆ UR (X);
(ii) LR (∅) = UR (∅) = ∅ and LR (U ) = UR (U ) = U ;
(iii) UR (X ∪ Y ) = UR (X) ∪ UR (Y );
(iv) UR (X ∩ Y ) ⊆ UR (X) ∩ UR (Y );
(v) LR (X ∪ Y ) ⊇ LR (X) ∪ LR (Y );
(vi) LR (X ∩ Y ) ⊆ LR (X) ∩ LR (Y );
(vii) LR (X) ⊆ LR (Y ) and UR (X) ⊆ UR (Y ) whenever X ⊆ Y ;
(viii) UR (X c ) = [LR (X)]c and LR (X c ) = [UR (X)]c ;
(ix) UR UR (X) = LR UR (X) = UR (X);
(x) LR LR (X) = UR LR (X) = LR (X).
Definition 2.2. [4]
Let U be the universe, R be an equivalence relation on U and τR (X) = {U, ∅, LR (X), UR (X),
BR (X)} where X ⊆ U. Then by the Property 2.2, τR (X) satisfies the following axioms:
(i) U and ∅ ∈ τR (X),
(ii) The union of the elements of any subcollection of τR (X) is in τR (X).,
(iii) The intersection of the elements of any finite subcollection of τR (X) is in τR (X)..
That is, τR (X) is a topology on U called the nano topology on U with respect to X. We call
(U, τR (X)) as the nano topological space. The elements of τR (X) are called as nano open sets
and [τR (X)]c is called as the dual nano topology of [τR (X)].
Remark 2.1. [4]
If [τR (X)] is the nano topology on U with respect to X, then the set B = {U, ∅, LR (X), BR (X)}
is the basis for τR (X).
Definition 2.3. [4]
If (U, τR (X)) is a nano topological space with respect to X and if A ⊆ U, then the nano
interior of A is defined as the union of all nano open subsets of A and it is denoted by N int(A).
That is, N int(A) is the largest nano open subset of A. The nano closure of A is defined as the
intersection of all nano closed sets containing A and it is denoted by N cl(A). That is, N cl(A)
is the smallest nano closed set containing A.
Definition 2.4. [4]
Let (U, τR (X)) be a nano topological space and A ⊆ U. Then A is said to be
(i) nano semi-open if A ⊆ N cl(N int(A))
(ii) nano pre open if A ⊆ N int(N cl(A))
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(iii) nano α-open if A ⊆ N int(N cl(N int(A)))
(iv) nano regular open if A = N int(N cl(A))
N SO(U, X), N P O(U, X) and N αO(U, X) respectively denote the families of all nano semiopen, nano pre open and nano α-open subsets of U.
Definition 2.5. [4]
(U, τR (X)) be a nano topological space and A ⊆ U. A is said to be nano semi-closed (resp.
nano pre closed, nano α-closed), if its complement is nano semi-open (nano pre open, nano
α-open).
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Nano β-open sets
Throughout this paper (U, τR (X)) is a nano topological space with respect to X where X ⊆ U,
R is an equivalence relation on U, U/R denotes the family of equivalence classes of U by R.
Definition 3.1.
A Subset A of a nano toplogical space (U, τR (X)) is nano β-open in U if A ⊆ N cl(N int(N cl(A))).
The set of all nano β-open sets of U is denoted by N βO(U, X).
Proposition 3.1.
If UR (X) = U in a nano topological space then N βO(U, X) is P (U ).
Proof
Let UR (X) = U.
(i) If LR (X) = ∅, then BR (X) = U. And then τR (X) = {U, ∅}. Thus for any set A,
N cl(N int(N cl(A))) = U. Hence A ⊆ N cl(N int(N cl(A))). Therefore A is nano β-open in
U. Hence N βO(U, X) is P (U ).
(ii) If LR (X) 6= ∅. Each element of U is either in LR (X) or in BR (X), since BR (X) =
UR (X) − LR (X).
If A ⊆ LR (X) then N cl(N int(N cl(A))) = LR (X) and if A ⊆ BR (X) then N cl(N int(N cl(A))) =
BR (X) because [LR (X)]c = BR (X), [BR (X)]c = LR (X) and UR (X) = U.. If A intersects both LR (X) and BR (X), N cl(A) = U. Then N cl(N int(N cl(A))) = U. Hence A ⊆
N cl(N int(N cl(A))). Therefore A is nano β-open in U. Hence N βO(U, X) is P (U ).
Proposition 3.2.
If UR (X) 6= U then U, ∅ and any set which intersects UR (X) are nano β-open set in U.
Proof
Case(i) LR (X) = ∅ or LR (X) = UR (X). In both situations τR (X) = {U, ∅, UR (X)}. If A intersects
UR (X) then N cl(A) = U. And hence N cl(N int(N cl(A))) = U. Therefore A is nano β-open
set in U. If A ⊆ [UR (X)]c , then N cl(A) = [UR (X)]c . And hence N cl(N int(N cl(A))) = ∅.
Therefore A is not nano β-open in U.
Case(ii) If LR (X) 6= ∅ then τR (X) = {U, ∅, LR (X), UR (X), BR (X)}.
1. If A ⊆ UR (X)
Suppose A ⊆ LR (X), then N cl(A) = [BR (X)]c and N int(N cl(A)) = LR (X) and hence
N cl(N int(N cl(A))) = [BR (X)]c ⊇ LR (X) ⊇ A. Therefore A is nano β-open in U. Suppose A ⊆ BR (X), then N cl(A) = [LR (X)]c and N int(N cl(A)) = BR (X) and hence
N cl(N int(N cl(A))) = [LR (X)]c ⊇ BR (X) ⊇ A. Therefore A is nano β-open in U. Suppose
A intersects both LR (X) and BR (X), N cl(A) = U and hence N cl(N int(N cl(A))) = U.
Therefore A is nano β-open in U.
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2. If A ⊆ [UR (X)]c , N cl(A) = [UR (X)]c . And hence N cl(N int(N cl(A))) = ∅. Therefore A
is not nano β-open in U.
3. If A intersects both UR (X) and [UR (X)]c .
Suppose A contains elements of LR (X) and [UR (X)]c , N cl(N int(N cl(A))) = [BR (X)]c =
[UR (X)]c ∪ LR (X) ⊇ A. Therefore A is nano β-open in U. Suppose A contains elements of
BR (X) and [UR (X)]c , N cl(N int(N cl(A))) = [LR (X)]c = [UR (X)]c ∪ BR (X) ⊇ A. Therefore A
is nano β-open in U. Suppose A intersects LR (X), BR (X) and [UR (X)]c then N cl(A) = U and
hence N cl(N int(N cl(A))) = U. Therefore A is nano β-open in U
Proposition 3.3.
If A is nano open in (U, τR (X)), then it is nano β-open in U.
Proof
Since A is nano open in U, N int(A) = A. And A ⊆ N cl(A) always. Then A ⊆ N cl(N int(A)) ⊆
N cl(N int(N cl(A))). Therefore A ⊆ N cl(N int(N cl(A))) Hence A is nano β-open in U.
Proposition 3.4.
Every nano semi-open set is nano β-open in (U, τR (X)).
Proof
If A is nano semi-open, then A ⊆ N cl(N int(A)) ⊆ N cl(N int(N cl(A))). Hence A is nano
β-open.
Proposition 3.5.
Every nano pre open set is nano β-open in (U, τR (X)).
Proof
If A is nano pre open, then A ⊆ N int(N cl(A)) ⊆ N cl(N int(N cl(A))). Hence A is nano
β-open.
Proposition 3.6.
Every nano regular open set is nano βopen in (U, τR (X)).
Proof
If A is nano regular open, then A = N int(N cl(A)) ⊆ N cl(N int(N cl(A))). Hence A is
nano β-open in U.
Proposition 3.7.
Every nano α-open set is nano β-open in (U, τR (X)).
Proof
Since N int(A) ⊆ A, N cl(N int(A)) ⊆ N cl(A). Therefore N int(N cl(N int(A))) ⊆ N int(N cl(A))
⊆ N cl(N int(N cl(A))). Hence A is nano β-open in U.
Remark 3.1.
The converse of the above proposition 3.3, 3.4, 3.5, 3.6 and 3.7 need not be true always as
seen from the following example.
Example 3.1.
Let U = {a, b, c, d}, U/R={{a, b}, {c}, {d}} and X = {a, c} Then LR (X) = {c}, UR (X) =
{a, b, c}, BR (X) = {a, b}. Therefore τR (X) = {U, ∅, {c}, {a, b, c}, {a, b}}. Now A = {a, d} is
nano β-open in U but not nano open, nano semi-open, nano pre open, nano regular open and
nano α-open in U.
Proposition 3.8.
If A and B are nano β-open sets in a space (U, τR (X)) then A ∪ B is also nano β-open in
U.
Proof
Since A and B are nano β-open sets, A ⊆ N cl(N int(N cl(A))) and B ⊆ N cl(N int(N cl(B))).
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Then A∪B ⊆ N cl(N int(N cl(A)))∪N cl(N int(N cl(B))) = N cl[N int(N cl(A))∪N int(N cl(B))] ⊆
N cl[N int[N cl(A) ∪ N cl(B)]]. Therefore A ∪ B ⊆ N cl(N int(N cl(A ∪ B))). Hence A ∪ B is also
nano β-open in U.
Remark 3.2.
The intersection of two nano β-open sets need not be nano β-open set. In Example 3.1, the
sets {a, d} and {b, d} are nano β-open sets in U but their intersection {d} is not nano β-open
in U.
Remark 3.3.
N βO(U, X) is supra topology on U.
Proposition 3.9.
N SO(U, X) ∪ N P O(U, X) ⊂ N βO(U, X)
Proof
Proof follows from proposition 3.4, 3.5 and 3.8.
Remark 3.4.
The equality in the above theorem does not hold in general. In Example 3.1, the set
A = {a, d} is nano β-open but not in N SO(U, X) ∪ N P O(U, X).
Proposition 3.10.
If V is nano open and A is nano β-open then V ∩ A is nano β-open.
Proof
V ∩ A ⊆ V ∩ N cl(N int(N cl(A))) ⊆ N cl(V ∩ N int(N cl(A))) ⊆ N cl(N int(N cl(V ∩ A))).
Therefore V ∩ A is nano β-open.
Proposition 3.11.
If B is nano subset of U and A is nano pre open in U such that A ⊆ B ⊆ N cl(N int(A)).
Then B is nano β-open in U.
Proof
Since A is nano preopen in U, A ⊆ N int(N cl(A)). Now B ⊆ N cl(N int(A)) ⊆ N cl(N int
(N int(N cl(A)))) = N cl(N int(N cl(A))) ⊆ N cl(N int(N cl(B))). Hence B ⊆ N cl(N int(N cl(B))).
Therefore B is nano β-open in U.
Proposition 3.12.
Each nano β-open set which is nano semi-closed is nano semi-open.
Proof
Let A ∈ N βO(U, X)and ∈ N SC(U, X). Then A ⊆ N cl(N int(N cl(A))) and N int(N cl(A)) ⊆
A. Hence N int(N cl(A)) ⊆ A ⊆ N cl(N int(N cl(A))). Since N int(N cl(A)) = G is a nano open
set in (U, τR (X)), i.e., there exists a nano open set such that G ⊆ A ⊆ N cl(G). Therefore A is
a nano semi-open set.
Proposition 3.13.
Each nano β-open set which is nano α-closed is nano closed.
Proof
Obvious.
References
[1] Abd EL-Monsef M. E., EL-Deep S. N. and Mahmoud R. A., B-Open Sets and B-continuous
mappings, Bull. Fac. Sci., Assiut Univ., 12 (1983), 77-90.
[2] Andrijevic D., Semi-pre open sets, Mat. Vesnik , 38 (1986), 24-32.
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[3] Ganster M. and Andrijevic D., On some questions concerning semi-preopen sets, J. Inst.
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[4] Lellis Thivagar M. and Carmel Richard, On Nano forms of Weakly open sets, International
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